Finite Group DatabasebetaFinite Group Databasebeta| order | 37278 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27159091 |
| Fitting, socle, derived | C3×C19×C109 ⋊ C2×C3 |
| Hall | C3×#20261880 ⋊ C2 |
| Hall | C19×C109 ⋊ C3×D3 |
| Sylow | C109 ⋊ D3×C19⋊C3 |
| Sylow | C19 ⋊ C3⋊(D109⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C19×C109 | C2×C3 |
| LCS | G ▷ C3×C19×C109 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 6213 |
A minimal faithful permutation representation — 2 generators on 131 points:
(2,3)(5,6,8)(7,10,11)(9,13,12)(14,18,16)(15,19,17)(20,22,21)(24,28,29,25,26,30)(27,36,37,31,32,46)(33,49,50,39,40,63)(34,48,44,42,38,35)(41,68,69,52,53,86)(43,66,67,55,51,60)(45,47,65,57,58,61)(54,91,92,71,72,113)(56,89,90,74,70,83)(59,64,88,76,77,84)(62,102,100,80,81,79)(73,118,119,94,95,131)(75,116,117,96,93,110)(78,87,115,97,98,111)(82,123,124,103,104,106)(85,127,121,107,108,101)(99,114,125,120,105,130)(109,129,122,112,128,126);(1,2,3)(4,5,7,11,16,21,18,6,9,14,19,8,12,17,10,15,20,22,13)(23,24,27,33,41,54,73,92,115,127,123,116,91,114,103,80,58,32,38,46,60,79,101,122,124,90,50,65,66,49,64,85,109,98,53,70,81,84,86,110,120,97,76,57,44,29,28,34,43,56,75,95,119,125,128,129,130,131,118,96,74,55,42,26,30,35,45,59,78,99,117,69,88,102,89,68,87,112,107,77,40,51,61,63,83,106,126,121,100,67,37,48,36,47,62,82,105,72,93,104,108,111,113,94,71,52,39,31,25)Polycyclic presentation:
2,3^2,19,109|c^a=c^2,e^a=e^108,d^b=d^7,e^b=e^631053 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37278, 99);